Area of a Sector

GCSE Geometry circle sector
Practice this formula All formulas
\( A = \frac{\theta}{360^{\circ}}\, \pi r^2 \)

Statement

The formula for the area of a sector is:

\[ A = \frac{\theta}{360^{\circ}} \times \pi r^2 \]

Here, \(A\) is the area of the sector, \(\theta\) is the central angle in degrees, and \(r\) is the radius of the circle.

Why it’s true (short reason)

  • The full area of a circle is \(\pi r^2\).
  • A full turn is \(360^{\circ}\).
  • The sector with angle \(\theta\) represents a fraction \(\theta/360\) of the circle’s area.

Recipe (how to use it)

  1. Identify the radius \(r\) and the central angle \(\theta\).
  2. Write the formula \(A=\frac{\theta}{360}\pi r^2\).
  3. Substitute the known values.
  4. Calculate, keeping answers in terms of \(\pi\) if asked, or rounding if required.

Spotting it

  • Look for shaded “slice” or “sector” diagrams of a circle.
  • Phrases like “area of a sector”, “shaded area of circle”, or “fraction of circle” are indicators.
  • Sometimes the diameter is given — halve it to find the radius before using the formula.

Common pairings

  • Arc length: often appears alongside area of a sector in exam questions.
  • Perimeter of a sector: requires both arc length and two radii.
  • Fractions of a circle: the formula is essentially “fraction of circle area”.

Mini examples

  1. Given: \(r=7\), \(\theta=90^{\circ}\). Find: area of sector. Answer: \((90/360)\pi \times 49 = 49\pi/4.\)
  2. Given: \(r=3\), \(\theta=120^{\circ}\). Find: area. Answer: \((120/360)\pi \times 9 = 3\pi.\)

Pitfalls

  • Using diameter instead of radius: always check carefully.
  • Mixing arc length and area formulas: both use \(\theta/360\), but one multiplies by circumference, the other by area.
  • Radians vs degrees: at GCSE level, use degrees.
  • Rounding too early: keep \(\pi\) exact until the final step.

Exam strategy

  • Underline radius and angle values immediately in the question.
  • Check units: if radius is in cm, the area is in cm\(^2\).
  • If asked for perimeter of sector, don’t stop at area — add arc length and two radii.
  • Always state the formula before substituting — it secures method marks.

Summary

The area of a sector is found by taking the fraction of the circle’s area corresponding to the angle at the centre. Multiply that fraction by \(\pi r^2\). This formula is widely used for problems involving fractions of circles, shaded regions, and real-life applications like slices of pizza or wedges of a wheel.